A Verification Theorem for Optimal Stopping Problems with Expectation Constraints

Ankirchner, Stefan; Klein, Maike; Kruse, Thomas

doi:10.1007/s00245-017-9424-2

Cited by 17 publications

(25 citation statements)

References 13 publications

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“…We note that since the original circulation of this paper, the same elliptic PDE has been obtained independently in [1] in a direct analysis of the expectation-constrained optimal stopping problem. The authors place relatively more emphasis on further analysis of the expectation-constrained optimal stopping problem, which may complement a reading of this paper by providing, for instance, specific examples of various degenerate behavior in the problem.…”

Section: Introductionmentioning

confidence: 92%

Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

Miller¹

2017

SIAM J. Control Optim.

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We present a novel method for solving a class of time-inconsistent optimal stopping problems by reducing them to a family of standard stochastic optimal control problems. In particular, we convert an optimal stopping problem with a non-linear function of the expected stopping time in the objective into optimization over an auxiliary value function for a standard stochastic control problem with an additional state variable. This approach differs from the previous literature which primarily employs Lagrange multiplier methods or relies on exact solutions. In contrast, we characterize the auxiliary value function as the unique viscosity solution of a non-linear elliptic PDE which satisfies certain growth constraints and investigate basic regularity properties. We demonstrate a connection between optimal stopping times for the original problem and optimal controls of the auxiliary control problem. Finally, we discuss extensions to more general dynamics and time-inconsistencies, as well as potential connections to degenerate Monge-Ampere equations.

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Section: Introductionmentioning

confidence: 92%

Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

Miller¹

2017

SIAM J. Control Optim.

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“…A dividend problem under the constraint that the surplus process must be above a given fixed level in order for dividend payments to be admissible is studied in [36]; see also [32] where this problem is studied in a model which allows for capital injection. Optimal stopping under expectation constraints is studied in [3,9] while stochastic control under expectation constraints is studied in [44]. Distributionconstrained optimal stopping is studied in [8,10].…”

Section: Background and Related Literaturementioning

confidence: 99%

Moment constrained optimal dividends: precommitment \& consistent planning

Christensen,

Lindensjö

2019

Preprint

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A moment constraint that limits the number of dividends in the optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz' consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

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“…Recently, Ankirchner et al [1] and Miller [42] took a different approach to optimal stopping problems for diffusion processes with expectation constraints by transforming them to stochastic optimization problems with martingale controls. The former characterizes the value function in terms of a Hamilton-Jacobi-Bellman equation and obtains a verification theorem, while the latter analyzes the optimal stopping problem with first-moment constraint that is embedded in a time-inconsistent (unconstrained) stopping problem.…”

Section: Introductionmentioning

confidence: 99%

Optimal Stopping with Expectation Constraints

Bayraktar¹,

Ye²

2020

Preprint

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We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints in a concrete probability setting is equivalent to the constrained problem in weak formulation (optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space). Thus the value of the optimal stopping problem with expectation constraints is independent of the specific probabilistic setting. Using a martingale-problem formulation, we make an equivalent characterization of the probability class in the weak formulation, which implies that the value function of the constrained optimal stopping problem is upper semi-analytic. Then we exploit a measurable selection argument to establish a dynamic programming principle (DPP) in the weak formulation for the value of the optimal stopping problem with expectation constraints, in which the conditional expected integrals of constraint functions act as additional states.

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A Verification Theorem for Optimal Stopping Problems with Expectation Constraints

Cited by 17 publications

References 13 publications

Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

Nonlinear PDE Approach to Time-Inconsistent Optimal Stopping

Moment constrained optimal dividends: precommitment \& consistent planning

Optimal Stopping with Expectation Constraints

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